Spectra of a class of non-symmetric operators in Hilbert spaces with applications to singular differential operators

2019 ◽  
Vol 150 (4) ◽  
pp. 1769-1790
Author(s):  
Huaqing Sun ◽  
Bing Xie
Keyword(s):  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Homogeneous Equation ◽  
Fundamental Theory ◽  
Linearly Independent ◽  
Sturm Liouville ◽  
Symmetric Operators ◽  
Complex Valued

AbstractThis paper is concerned with a class of non-symmetric operators, that is, 𝒥-symmetric operators, in Hilbert spaces. A sufficient condition for λ ∈ C being an element of the essential spectrum of a 𝒥-symmetric operator is given in terms of the number of linearly independent solutions of a certain homogeneous equation, and a characterization for points of the essential spectrum plus the set of all eigenvalues of a 𝒥-symmetric operator is obtained in terms of the numbers of linearly independent solutions of certain inhomogeneous equations. As direct applications, the corresponding results are obtained for singular 𝒥-symmetric Hamiltonian systems and their special forms of singular Sturm-Liouville equations with complex-valued coefficients, which enable us to study the spectra of singular 𝒥-symmetric differential expressions using numerous tools available in the fundamental theory of differential equations.

On the location of the essential spectra and regularity fields of complex Sturm—Liouville operators

1980 ◽  
Vol 85 (1-2) ◽  
pp. 1-14 ◽  
Author(s):  
David Race
Keyword(s):  
Essential Spectrum ◽  
Essential Spectra ◽  
Linearly Independent ◽  
Complete Determination ◽  
Associated Operators ◽  
Sturm Liouville ◽  
Selfadjoint Extensions ◽  
Complex Valued ◽  
Liouville Operators

SynopsisIn this paper the Sturm-Liouville expression τy= −(py′)′ +qy, with complex-valued coefficients is considered, and a number of results concerning the location of the essential spectrum of associated operators are obtained. Some of these are extensions or generalizations of results due to Birman, and Glazman, whilst others are new. These lead to criteria for the non-emptiness of the regularity field of the corresponding minimal operator—a condition which is needed in the theory ofJ-selfadjoint extensions. A complete determination of the regularity field is made when the equation τy= λ0yhas two linearly independent solutions inL2[a,∞) for some complex λ0.

On an open problem of Weidmann: essential spectra and square-integrable solutions

2011 ◽  
Vol 141 (2) ◽  
pp. 417-430 ◽  
Author(s):  
Jiangang Qi ◽  
Shaozhu Chen
Keyword(s):  
Essential Spectrum ◽  
Open Problem ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Essential Spectra ◽  
Entire Interval ◽  
Linearly Independent ◽  
Adjoint Extension ◽  
Set Up ◽  
The Right

In 1987, Weidmann proved that, for a symmetric differential operator τ and a real λ, if there exist fewer square-integrable solutions of (τ−λ)y = 0 than needed and if there is a self-adjoint extension of τ such that λ is not its eigenvalue, then λ belongs to the essential spectrum of τ. However, he posed an open problem of whether the second condition is necessary and it has been conjectured that the second condition can be removed. In this paper, we first set up a formula of the dimensions of null spaces for a closed symmetric operator and its closed symmetric extension at a point outside the essential spectrum. We then establish a formula of the numbers of linearly independent square-integrable solutions on the left and the right subintervals, and on the entire interval for nth-order differential operators. The latter formula ascertains the above conjecture. These results are crucial in criteria of essential spectra in terms of the numbers of square-integrable solutions for real values of the spectral parameter.

The Expansion Theorems for Sturm-Liouville Operators with an Involution Perturbation

2021 ◽  
Author(s):  
Abdizhahan Sarsenbi
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Second Order ◽  
Order Differential Operator ◽  
Spectral Expansions ◽  
Sturm Liouville ◽  
Complex Valued ◽  
Liouville Operators

In this work, we studied the Green’s functions of the second order differential operators with involution. Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution is obtained. Basicity of eigenfunctions of the second-order differential operator operator with complex-valued coefficient is established.

An inverse problem for Sturm–Liouville operators on the half-line with complex weights

2019 ◽  
Vol 27 (3) ◽  
pp. 439-443
Author(s):  
Vjacheslav Yurko
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Operators ◽  
Spectral Characteristics ◽  
Half Line ◽  
Sturm Liouville ◽  
The Given ◽  
Complex Valued ◽  
The Uniqueness Theorem ◽  
Liouville Operators

Abstract Second order differential operators on the half-line with complex-valued weights are considered. Properties of spectral characteristics are established, and the inverse problem of recovering operator’s coefficients from the given Weyl-type function is studied. The uniqueness theorem is proved for this class of nonlinear inverse problems, and a number of examples are provided.

scholarly journals On the Solvability Complexity Index for Unbounded Selfadjoint and Schrödinger Operators

2019 ◽  
Vol 91 (6) ◽  
Author(s):  
Frank Rösler
Keyword(s):  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Selfadjoint Operator ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Selfadjoint Operators ◽  
Complexity Index ◽  
Compact Perturbations ◽  
Complex Valued

AbstractWe study the solvability complexity index (SCI) for unbounded selfadjoint operators on separable Hilbert spaces and perturbations thereof. In particular, we show that if the extended essential spectrum of a selfadjoint operator is convex, then the SCI for computing its spectrum is equal to 1. This result is then extended to relatively compact perturbations of such operators and applied to Schrödinger operators with (complex valued) potentials decaying at infinity to obtain $${\text {SCI}}=1$$SCI=1 in this case, as well.

Lower Bounds for the Essential Spectrum of Fourth-Order Differential Operators

1969 ◽  
Vol 21 ◽  
pp. 460-465
Author(s):  
Kurt Kreith
Keyword(s):  
Lower Bound ◽  
Essential Spectrum ◽  
Lower Bounds ◽  
Fourth Order ◽  
Differential Operators ◽  
Intimate Connection ◽  
Sturm Liouville ◽  
Image Position ◽  
Oscillation Properties ◽  
Liouville Operators

In this paper, we seek to determine the greatest lower bound of the essential spectrum of self-adjoint singular differential operators of the form1where 0 ≦ x < ∞. In the event that this bound is + ∞, our results will yield criteria for the discreteness of the spectrum of (1).Such bounds have been established by Friedrichs (3) for Sturm-Liouville operators of the formand our techniques will be closely related to those of (3). However, instead of studying the solutions of2directly, we shall exploit the intimate connection between the infimum of the essential spectrum of (1) and the oscillation properties of (2).

On the essential spectra of linear 2nth order differential operators with complex coefficients

1982 ◽  
Vol 92 (1-2) ◽  
pp. 65-75 ◽  
Author(s):  
David Race
Keyword(s):  
Differential Expression ◽  
Essential Spectrum ◽  
Differential Operators ◽  
Essential Spectra ◽  
Associated Operators ◽  
Linear Differential ◽  
Selfadjoint Extensions ◽  
Complex Valued ◽  
Complex Coefficients ◽  
Liouville Operators

SynopsisIn this paper, a formally J-symmetric, linear differential expression of 2nth order, with complex-valued coefficients, is considered. A number of results concerning the location of the essential spectrum of associated operators are obtained. These are extensions of earlier work dealing with complex Strum-Liouville operators, and include results which, in the real case, are due to Birman, Glazman and others. They lead to criteria, for the non-emptiness of the regularity field, of the corresponding minimal operator-a condition which is needed in the theory of J-selfadjoint extensions.

A general HELP inequality connected with symmetric operators

2005 ◽  
Vol 462 (2066) ◽  
pp. 587-606 ◽  
Author(s):  
Matthias Langer
Keyword(s):  
Hilbert Space ◽  
Symmetric Operator ◽  
Weyl Function ◽  
Difference Operators ◽  
Operator Matrix ◽  
Block Operator ◽  
Abstract Boundary ◽  
Sturm Liouville ◽  
Symmetric Operators ◽  
Liouville Operators

In this paper, a general HELP (Hardy–Everitt–Littlewood–Pólya) inequality is considered which is connected with a symmetric operator in a Hilbert space and abstract boundary mappings. A criterion for the validity of such an inequality in terms of the abstract Titchmarsh–Weyl function is proved and applied to Sturm–Liouville operators, difference operators, a Hamiltonian system and a block operator matrix.

On the exponential behaviour of eigenfunctions and the essential spectrum of differential operators

1982 ◽  
Vol 92 (3-4) ◽  
pp. 271-300
Author(s):  
F. V. Atkinson ◽  
W. D. Evans
Keyword(s):  
Differential Equation ◽  
Linear Operator ◽  
Essential Spectrum ◽  
General Result ◽  
Differential Operators ◽  
Exponential Behaviour ◽  
Image Position ◽  
Complex Valued

SynopsisThe paper deals with the differential equationon [ 0, ∞) Where λ>0 and the coefficients qm are complex-valued with qn continuous and non-zero, w is positive and continuous and qm for m = 0, 1,…, n − 1. In the first part of the paper the exponential behaviour of any solution of (*) is given in terms of a function ρ(λ) which is roughly the distance of λ from the essential spectrum of a closed, densely denned linear operator T generated by T+ in L2(0, ∞ w). Next, estimates are obtained for the solutions in terms of the coefficients in (*). When the latter results are compared with the estimates established previously in terms of ρ(λ), bounds for ρ(λ) are obtained. From the general result there are two kinds of consequences. In the first, criteria for ρ(λ) = 0 for all All λ > 0 are obtained; this means that [0, ∞) lies in the essential spectrum of T in appropriate circumstances. The second type of consequence concerns bounds of the form ρ(λ) = O(λr) for λ → ∞ and r<1.

Boundary conditions and reducibility of differential operators

1984 ◽  
Vol 96 (3) ◽  
pp. 549-553
Author(s):  
B. Fisherl ◽  
Z. Zahreddine
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Maximal Operator ◽  
Differential Operators ◽  
Deficiency Index ◽  
Linearly Independent ◽  
Symmetric Operators

AbstractExamples were exhibited in [4] of both reducible and irreducible symmetric operators (of deficiency index (1:1)) associated with − d2/dt2 in the Hilbert space L2(I) (I = [0,1). Such symmetric operators are determined by three linearly independent boundary conditions which define their domains as restrictions of the domain of the maximal operator associated with — d2/dt2.

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